Optimal. Leaf size=21 \[ -\frac {2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{d} \]
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Rubi [A] time = 0.02, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {3011} \[ -\frac {2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 3011
Rubi steps
\begin {align*} \int \sqrt {\cos (c+d x)} \left (3-5 \cos ^2(c+d x)\right ) \, dx &=-\frac {2 \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{d}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 23, normalized size = 1.10 \[ -\frac {\sin (2 (c+d x)) \sqrt {\cos (c+d x)}}{d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 19, normalized size = 0.90 \[ -\frac {2 \, \cos \left (d x + c\right )^{\frac {3}{2}} \sin \left (d x + c\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int -{\left (5 \, \cos \left (d x + c\right )^{2} - 3\right )} \sqrt {\cos \left (d x + c\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.62, size = 99, normalized size = 4.71 \[ -\frac {4 \sqrt {\left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}\, \sqrt {2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}}{\sin \left (\frac {d x}{2}+\frac {c}{2}\right ) d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\int {\left (5 \, \cos \left (d x + c\right )^{2} - 3\right )} \sqrt {\cos \left (d x + c\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.05 \[ -\int \sqrt {\cos \left (c+d\,x\right )}\,\left (5\,{\cos \left (c+d\,x\right )}^2-3\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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